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The Discover and Citi DC wil report the AU but you can't set limits. You can set a limit with Amex, but I don't know if this particular card can be set like that. Chase Slate does report, so I would think the Freedom would too. I have no idea about the other cards.
Costco Anywhere Visa can set an AU's spending limit... think its the only Citi card that can
https://www.nerdwallet.com/blog/credit-cards/credit-card-spending-limit-for-authorized-users/
If your key objective is to help your GF repair her credit, there are some cards that allow you to add an AU without even issuing a card, or have the card shipped to you where you could put it away. In that case, you wouldn't need to worry about setting a spending limit.
Is there a list of cards that can add an AU without issuing a card?